Galaxy Luminosity Function of the Coma Cluster from Deep $u'-g'-r'$ Wendelstein Imaging Data

We derive the $g'$ band galaxy luminosity function (GLF) of quenched red sequence in the Coma cluster from a deep-imaging survey with $\approx1.5\,\mathrm{deg^2}$ around the cluster center. The dataset comprises deep $u'$-, $g'$-, and $r'$-band data obtained with the Wendelstein Wide Field Imager on the 2.1\,m Fraunhofer Wendelstein Telescope reaching median $3\sigma$ surface brightness limits in $10"\times10"$ boxes of $\mathrm{ (30.0\,u',\,\,29.6\,g',\,\,28.7\,r')\,mag\,arcsec^{-2}}$. We measure structural parameters across a large dynamic range in galaxy brightness ($-24.5\,g'\,\mathrm{mag} \lessapprox M\lessapprox-11.3\,g'\,\mathrm{mag}$), from the brightest cluster galaxy to low-luminosity dwarfs, including compact dwarf galaxies and ultra-diffuse galaxies. We automatically identify 5161 cluster member candidates based on their membership on the quiescent sequence in the $u'-g'$ versus $g'-r'$ color--color diagram and their red sequence membership. The structural parameters of bright galaxies are obtained via isophotal modeling, and fully automated parametric image fitting for faint ones. Injection-recovery tests and two identically analyzed reference fields provide statistical corrections for completeness and contamination, yielding a representative GLF that reliably probes the faint end and may serve as a benchmark for future studies. We report a best-fit double Schechter $g'$ band GLF with a comparatively steep faint-end slope $\alpha_2=-1.539^{+0.024}_{-0.024}$. We perform an apples-to-apples comparison with the Coma counterpart in the SLOW simulations, finding overall agreement but a deficit of bright and an excess of dwarf galaxies in the simulation, despite its shallower faint-end slope, highlighting the need for true apples-to-apples comparisons that consider both normalization and slope for a consistent interpretation of the GLF.